The next-to-minimal weights of binary projective Reed-Muller codes

TL;DR

This paper determines the next-to-minimal weights of binary projective Reed-Muller codes, filling a gap in understanding their higher Hamming weights and code performance.

Contribution

It provides a complete characterization of the next-to-minimal weights for these codes, extending prior knowledge limited to minimum distances.

Findings

Next-to-minimal weights are fully determined for binary projective Reed-Muller codes.

In most cases, these weights match those of Reed-Muller codes.

The results reveal exceptions where weights differ from Reed-Muller codes.

Abstract

Projective Reed-Muller codes were introduced by Lachaud, in 1988 and their dimension and minimum distance were determined by Serre and S{\o}rensen in 1991. In coding theory one is also interested in the higher Hamming weights, to study the code performance. Yet, not many values of the higher Hamming weights are known for these codes, not even the second lowest weight (also known as next-to-minimal weight) is completely determined. In this paper we determine all the values of the next-to-minimal weight for the binary projective Reed-Muller codes, which we show to be equal to the next-to-minimal weight of Reed-Muller codes in most, but not all, cases.

Equations

$$W_{\text{RM}}^{(1)}(n,d)=(q-b)q^{n-a-1}.$$

$$W_{\text{RM}}^{(2)}(n,d)=(q-b)q^{n-a-1}+c{q}^{n-a-2}=\left(1+\frac{c}{(q-b)q}\right)(q-b)q^{n-a-1}$$

$$W_{\text{PRM}}^{(1)}(n,d)=W_{\text{RM}}^{(1)}(n,d-1).$$

$$W_{\text{PRM}}^{(2)}(n,d)\le W_{\text{RM}}^{(2)}(n,d-1).$$

$$d-1=k(q-1)+\ell$$

$$W_{\text{PRM}}^{(1)}(r,d)=W_{\text{RM}}^{(1)}(r,d-1)=\{\begin{array}{cc}1& \text{if }0\le r\le k;\\ (q-\ell )q^{r-k-1}& \text{if }k<r\le n.\end{array}$$

$$|S|<\left(1+\frac{1}{q}\right)(q-\ell )q^{n-k-1}=\left(q-\ell +1-\frac{\ell }{q}\right)q^{n-k-1}<q^{n-k}\le q^n<|{\mathbb{P}}^n({\mathbb{F}}_q)|,$$

$${\mathcal{G}}_F=\{G\text{ a linear subspace of }{\mathbb{P}}^n({\mathbb{F}}_q)\mid F\subset G\text{ and }\dim G=r+1\}.$$

$$|{\mathcal{G}}_F|\left(\frac{q^{r+2}-1}{q-1}-\frac{q^{r+1}-1}{q-1}\right)+\frac{q^{r+1}-1}{q-1}=\frac{q^{n+1}-1}{q-1}$$

$$|S|=\sum _{G\in {\mathcal{G}}_F}|S\cap G|\ge \frac{q^{n-r}-1}{q-1}\cdot W_{\text{PRM}}^{(1)}(r+1,d).$$

$$\frac{q^{n-k+1}-1}{q-1}=q^{n-k}+\cdots +q+1<\left(q-\ell +1-\frac{\ell }{q}\right)q^{n-k-1}\le q^{n-k}-q^{n-k-2}$$

$$\frac{q^{n-r}-1}{q-1}(q-\ell )q^{r-k}<\left(1+\frac{1}{q}\right)(q-\ell )q^{n-k-1},$$

$$|S|\le (q-\ell +1)q^{n-k-1}\le q^{n-k}\le q^n<|{\mathbb{P}}^n({\mathbb{F}}_q)|,$$

$${\mathcal{G}}_F=\{G\text{ a linear subspace of }{\mathbb{P}}^n({\mathbb{F}}_q)\mid F\subset G\text{ and }\dim G=r+1\},$$

$$|S|=\sum _{G\in {\mathcal{G}}_F}|S\cap G|\ge \frac{q^{n-r}-1}{q-1}\cdot W_{\text{PRM}}^{(1)}(r+1,d).$$

$$\frac{q^{n-r}-1}{q-1}=q^{n-r-1}+\cdots +1\le (q-\ell +1)q^{n-k-1}=q^{n-k}-(\ell -1)q^{n-k-1}\le q^{n-r-1}$$

$$W^{(2)}_{\textrm{RM}}(n,d-1)=\left\{\begin{array}[]{ll}2^{n}&\text{ if \ }k=0\\ 3\cdot 2^{n-k-2}&\text{ if \ }0<k<n-2\\ 4&\text{ if \ }k=n-2.\\ \end{array}\right.$$

$$W_{\text{PRM}}^{(2)}(n,d)=W_{\text{RM}}^{(2)}(n,d-1),$$

$$W_{\text{PRM}}^{(2)}(n,d)=3\cdot 2^{n-2}<W_{\text{RM}}^{(2)}(n,d-1).$$

$$0\ne |f|<\left(1+\frac{1}{2}\right)W_{\text{PRM}}^{(1)}(n,d)$$

$$\left(1+\frac{1}{2}\right)W_{\text{PRM}}^{(1)}(n,d)=3\cdot 2^{n-2}<W_{\text{RM}}^{(2)}(n,d-1)=2^n.$$

$$3\cdot 2^{n-2}\le W_{\text{PRM}}^{(2)}(n,d),$$

$$0\ne |f|<\left(1+\frac{1}{2}\right)2^{n-k-1}$$

$$g(Q_2):=f_2(1,1,Q_2)+f_1(1,Q_2)+f_0(1,Q_2)=1.$$

$$|f_2(1,1,X_2,\dots ,X_n)|=W_{\text{RM}}^{(1)}(n-1,n-2)=2$$

$$|f_2(1,1,X_2,\dots ,X_n)|\ge W_{\text{RM}}^{(2)}(n-1,n-2)=4.$$

Full text

The next-to-minimal weights of binary projective Reed-Muller codes

Cícero Carvalho and V. G. Lopez Neumann111Authors’ emails: cicero@ufu.br and victor.neumann@ufu.br. Both authors were partially supported by grants from CNPq and FAPEMIG.

Published in IEEE Transactions on Information Theory, vol. 62, issue 11, Nov. 2016.

http://doi.org/10.1109/TIT.2016.2611527

Faculdade de Matemática, Universidade Federal de Uberlândia, Av. J. N. Ávila 2121, 38.408-902   Uberlândia - MG, Brazil

Abstract. Projective Reed-Muller codes were introduced by Lachaud, in 1988 and their dimension and minimum distance were determined by Serre and Sørensen in 1991. In coding theory one is also interested in the higher Hamming weights, to study the code performance. Yet, not many values of the higher Hamming weights are known for these codes, not even the second lowest weight (also known as next-to-minimal weight) is completely determined. In this paper we determine all the values of the next-to-minimal weight for the binary projective Reed-Muller codes, which we show to be equal to the next-to-minimal weight of Reed-Muller codes in most, but not all, cases.

1 Introduction

The so-called Reed-Muller codes appeared in 1954, when they were defined by D.E. Muller ([9]) and given a decoding algorithm by I.S. Reed ([10]). They were codes defined over $\mathbb{F}_{2}$ and in 1968 Kasami, Lin, and Peterson ([5]) extended the original definition to a finite field $\mathbb{F}_{q}$, where $q$ is any prime power. They named these codes “generalized Reed-Muller codes” and presented some results on the weight distribution, the dimension of the codes being determined in later works. In coding theory one is always interested in the values of the higher Hamming weights of a code because of their relationship with the code performance, but usually this is not a simple problem. For the generalized Reed-Muller codes, the complete determination of the second lowest Hamming weight, also called next-to-minimal weight, was only completed in 2010, when Bruen ([1]) observed that the value of these weights could be obtained from unpublished results in the Ph.D. thesis of D. Erickson ([3]) and Bruen’s own results from 1992 and 2006.

Twenty years after the definition of the generalized Reed-Muller codes the class of projective Reed-Muller codes was introduced by Lachaud ([6]). The parameters of these codes were determined by Serre ([16]), for some cases, and by Sørensen ([17]) for the general case, and they proved that the minimum distance of the projective Reed-Muller codes of order $d$ is equal to the minimum distance of the generalized Reed-Muller code of order $d-1$ (see (2.1)). The determination of the next-to-minimal weight for these codes is yet to be done, and there are some results (also about higher Hamming weights) on this subject by Rodier and Sboui ([12], [13], [15]) and also by Ballet and Rolland ([2]). In this paper we present all the values of the next-to-minimal weights for the case of binary projective Reed-Muller codes. Interestingly, we note that the next-to-minimal weight of the binary projective Reed-Muller codes of order $d$ is equal to the next-to-minimal weight of the Reed-Muller codes of order $d-1$ in most but not all cases (see Theorem 3.1). In the next section we recall the definitions of the generalized and projective Reed-Muller codes, and prove some results of geometrical nature that will allow us to determine the next-to-minimal weight of the binary projective Reed-Muller codes, which is done in the last section.

2 Preliminary results

Let $\mathbb{F}_{q}$ be a finite field and let $I_{q}=(X_{1}^{q}-X_{1},\ldots,X_{n}^{q}-X_{n})\subset\mathbb{F}_{q}[X_{1},\ldots,X_{n}]$ be the ideal of polynomials which vanish at all points $P_{1},\ldots,P_{q^{n}}$ of the affine space $\mathbb{A}^{n}(\mathbb{F}_{q})$. Let $\varphi:\mathbb{F}_{q}[X_{1},\ldots,X_{n}]/I_{q}\rightarrow\mathbb{F}_{q}^{q^{n}}$ be the $\mathbb{F}_{q}$-linear transformation given by $\varphi(g+I_{q})=(g(P_{1}),\ldots,g(P_{q^{n}}))$.

Definition 2.1

Let $d$ be a nonnegative integer. The generalized Reed-Muller code of order $d$ is defined as $\textrm{RM}(n,d)=\{\varphi(g+I_{q})\,|\,g=0\textrm{ or }\deg(g)\leq d\}$.

One may show that $\textrm{RM}(n,d)=\mathbb{F}_{q}^{q^{n}}$ if $d\geq n(q-1)$, so in this case the minimum distance is 1. Let $d\leq n(q-1)$ and write $d=a(q-1)+b$ with $0<b\leq q-1$, then the minimum distance of $\textrm{RM}(n,d)$ is

[TABLE]

The next-to-minimal weight $W^{(2)}_{\textrm{RM}}(n,d)$ of $\textrm{RM}(n,d)$ is equal to

[TABLE]

where $c$ is equal to $b-1$, $q-1$ or $q$, according to the values of $q$ and $d$ (see [2, Theorems 9 and 10]). We will quote specific values of $W^{(2)}_{\textrm{RM}}(n,d)$ when we need them.

Let $Q_{1},\ldots,Q_{N}$ be the points of $\mathbb{P}^{n}(\mathbb{F}_{q})$, where $N=q^{n}+\ldots+q+1$. It is known (see e.g. [8] or [11]) that the homogeneous ideal $J_{q}\subset\mathbb{F}_{q}[X_{0},\ldots,X_{n}]$ of the polynomials which vanish in all points of $\mathbb{P}^{n}(\mathbb{F}_{q})$ is generated by $\{X_{j}^{q}X_{i}-X_{i}^{q}X_{j}\,|\,0\leq i<j\leq n\}$. We denote by $\mathbb{F}_{q}[X_{0},\ldots,X_{n}]_{d}$ (respectively, $(J_{q})_{d}$) the $\mathbb{F}_{q}$-vector subspace formed by the homogeneous polynomials of degree $d$ (together with the zero polynomial) in $\mathbb{F}_{q}[X_{0},\ldots,X_{n}]$ (respectively, $J_{q}$).

Definition 2.2

Let $d$ be a positive integer and let $\psi:\mathbb{F}_{q}[X_{0},\ldots,X_{n}]_{d}/(J_{q})_{d}\rightarrow\mathbb{F}_{q}^{N}$ be the $\mathbb{F}_{q}$-linear transformation given by $\psi(f+(J_{q})_{d})=(f(Q_{1})\ldots,f(Q_{n}))$, where we write the points of $\mathbb{P}^{n}(\mathbb{F}_{q})$ in the standard notation, i.e. the first nonzero entry from the left is equal to 1. The projective Reed-Muller code of order $d$, denoted by $\textrm{PRM}(n,d)$, is the image of $\psi$.

The minimum distance $W^{(1)}_{\textrm{PRM}}(n,d)$ of $\textrm{PRM}(n,d)$ was determined by Serre ([16]) and Sørensen (see [17]) who proved that

[TABLE]

Let $\omega$ be the Hamming weight of $\varphi(g+I_{q})$, where $g\in\mathbb{F}_{q}[X_{1},\ldots,X_{n}]$ is a polynomial of degree $d-1$, and let $g^{(h)}$ be the homogenization of $g$ with respect to $X_{0}$. Then the degree of $g^{(h)}$ is $d-1$ and the weight of $\psi(X_{0}g^{(h)}+(J_{q})_{d})$ is $\omega$. This shows that, denoting by $W^{(2)}_{\textrm{PRM}}(n,d)$ the next-to-minimal weight of $\textrm{PRM}(n,d)$, we have

[TABLE]

In the next section we will prove that, for binary projective Reed-Muller codes, equality holds in most but not all cases (see Theorem 3.1).

Definition 2.3

Let $f\in\mathbb{F}_{q}[X_{0},\ldots,X_{n}]_{d}$. The set of points of $\mathbb{P}^{n}(\mathbb{F}_{q})$ which are not zeros of $f$ is called the support of $f$, and we denote its cardinality by $|f|$ (hence $|f|$ is the weight of the codeword $\psi(f+(J_{q})_{d})$).

Lemma 2.4

*Let $f\in\mathbb{F}_{q}[X_{0},\ldots,X_{n}]_{d}$ be a nonzero polynomial, and let $S$ be it support. Let $G\subset\mathbb{P}^{n}(\mathbb{F}_{q})$ be a linear subspace of dimension $r$, with $r\in\{1,\ldots,n-1\}$, then either $S\cap G=\emptyset$ or $|S\cap G|\geq W^{(1)}_{\textrm{PRM}}(r,d)$. *

**Proof: ** After a projective transformation we may assume that $G$ is given by $X_{r+1}=\cdots=X_{n}=0$. Assume that $|S\cap G|\neq\emptyset$ and let $g$ be the polynomial obtained from $f$ by evaluating $X_{i}=0$ for $i=r+1,\ldots,n$. Then $g$ is a nonzero homogeneous polynomial of degree $d$ and its support is equal to $S\cap G$. Considering $g$ as a polynomial which evaluates at points of $\mathbb{P}^{r}(\mathbb{F}_{q})$ we have $|S\cap G|=|g|\geq W^{(1)}_{\textrm{PRM}}(r,d)$. $\Box$

Observe that when $d=1$ we do not have a next-to-minimal weight for $\textrm{PRM}(n,1)$ since all hyperplanes in $\mathbb{P}^{n}(\mathbb{F}_{q})$ have the same number of zeros. In [17, Remark 3] Sørensen proved that $\textrm{PRM}(n,d)=\mathbb{F}_{q}^{N}$ whenever $d\geq n(q-1)+1$, so from now on we assume that $2\leq d\leq n(q-1)$.

Lemma 2.5

Let $f\in\mathbb{F}_{q}[X_{0},\ldots,X_{n}]_{d}$ be a polynomial with a nonempty support $S$. If there exists a hyperplane $H\subset\mathbb{P}^{n}(\mathbb{F}_{q})$ such that $S\cap H=\emptyset$ and $|f|>W^{(1)}_{\textrm{PRM}}(n,d)$ then $|f|\geq W^{(2)}_{\textrm{RM}}(n,d-1)$.

**Proof: ** After a projective transformation we may assume that $H$ is the hyperplane defined by $X_{0}=0$. Writing $f=X_{0}f_{1}+f_{2}$, where $f_{1}\in\mathbb{F}_{q}[X_{0},\ldots,X_{n}]_{d-1}$ and $f_{2}\in\mathbb{F}_{q}[X_{1},\ldots,X_{n}]_{d}$, from $S\cap H=\emptyset$ we get that $f_{2}$ vanishes on $H$ and a fortiori on $\mathbb{P}^{n}(\mathbb{F}_{q})$, so $f_{2}\in J_{q}$. Let $g$ be the polynomial obtained from $f_{1}$ by evaluating $X_{0}=1$, then $g$ is not zero (otherwise $S=\emptyset$) and $\deg(g)\leq d-1$. Considering $g$ as a polynomial which evaluates at $\mathbb{A}^{n}(\mathbb{F}_{q})$ we see that the number $|g|$ of points where $g$ is not zero is equal to $|f|$. Since $|f|>W^{(1)}_{\textrm{PRM}}(n,d)=W^{(1)}_{\textrm{RM}}(n,d-1)$ we must have $|g|\geq W^{(2)}_{\textrm{RM}}(n,d-1)$. $\Box$

In what follows the integers $k$ and $\ell$ will always be the ones uniquely defined by the equality

[TABLE]

with $0\leq k\leq n-1$ and $0<\ell\leq q-1$. Then, from the data on the minimum distance of $\textrm{RM}(n,d)$ we get, for $0\leq r\leq n$, that

[TABLE]

Proposition 2.6

Let $S\subset\mathbb{P}^{n}(\mathbb{F}_{q})$ be a nonempty set and assume that $S$ has the following properties:

$|S|<\left(1+\dfrac{1}{q}\right)(q-\ell)q^{n-k-1}$. 2. 2.

For every linear subspace $G\subset\mathbb{P}^{n}(\mathbb{F}_{q})$ of dimension $s$, with $s\in\{1,\ldots,n-1\}$, either $S\cap G=\emptyset$ or $|S\cap G|\geq W^{(1)}_{\textrm{PRM}}(s,d)$.

Then there exists a hyperplane $H\subset\mathbb{P}^{n}(\mathbb{F}_{q})$ such that $S\cap H=\emptyset$.

**Proof: ** We start by noting that

[TABLE]

so let $0\leq r<n$ be the largest integer such that there is a linear subspace $F\subset\mathbb{P}^{n}(\mathbb{F}_{q})$ of dimension $r$ satisfying $S\cap F=\emptyset$, we want to show that $r=n-1$. Let

[TABLE]

The intersection of two distinct elements of $\mathcal{G}_{F}$ is $F$, $|G|=(q^{r+2}-1)/(q-1)$ for all $G\in\mathcal{G}_{F}$ and any point of $\mathbb{P}^{n}(\mathbb{F}_{q})$ outside $F$ belongs to some $G\in\mathcal{G}_{F}$ hence

[TABLE]

and we get $|\mathcal{G}_{F}|=(q^{n-r}-1)/(q-1)$. From $S=\bigcup_{G\in\mathcal{G}_{F}}(S\cap G)$, $S\cap F=\emptyset$ and property 2 we get

[TABLE]

Assume that $r<k$, then $W^{(1)}_{\textrm{PRM}}(r+1,d)=1$ and from property 1 we get $(q^{n-r}-1)/(q-1)<(1+1/q)(q-\ell)q^{n-k-1}$. Since the left-hand side decreases with $r$ we plug in $r=k-1$ and get

[TABLE]

which is absurd. Now we assume that $k\leq r\leq n-1$, and again from property 1 we get

[TABLE]

hence $q^{n-r}-1<q^{n-r}-q^{n-r-2}$ which is only possible when $r=n-1$. $\Box$

Proposition 2.7

Let $S\subset\mathbb{P}^{n}(\mathbb{F}_{q})$ be a nonempty set and assume that $S$ has the following properties:

$|S|\leq\left(1+\dfrac{1}{(q-\ell)}\right)(q-\ell)q^{n-k-1}=(q-\ell+1)q^{n-k-1}$. 2. 2.

For every linear subspace $G\subset\mathbb{P}^{n}(\mathbb{F}_{q})$ of dimension $s$, with $s\in\{1,\ldots,n-1\}$, either $S\cap G=\emptyset$ or $|S\cap G|\geq W^{(1)}_{\textrm{PRM}}(s,d)$.

Then there exists $r\geq k$ and a linear subspace $H_{r}\subset\mathbb{P}^{n}(\mathbb{F}_{q})$ of dimension $r$ such that $S\cap H_{r}=\emptyset$.

**Proof: ** We start as in the proof of the previous Lemma, and observe that

[TABLE]

so let $0\leq r<n$ be the largest integer such that there is a linear subspace $F\subset\mathbb{P}^{n}(\mathbb{F}_{q})$ of dimension $r$ satisfying $S\cap F=\emptyset$. Let

[TABLE]

as before we have $|\mathcal{G}_{F}|=(q^{n-r}-1)/(q-1)$ and

[TABLE]

Assume that $r<k$, then $W^{(1)}_{\textrm{PRM}}(r+1,d)=1$ and from property 1 we get

[TABLE]

which is absurd, so we must have $k\leq r\leq n-1$. $\Box$

3 Main results

In this section we determine the next-to-minimal weight for the binary projective Reed-Muller codes. Recall that we are assuming that $2\leq d\leq n(q-1)$ so if $q=2$ we have $n\geq 2$. Also, from $d-1=k(q-1)+\ell$, with $0\leq k\leq n-1$ and $0<\ell\leq q-1$ we see that when $q=2$ we have $\ell=1$ and $d-1=k+1$, so from $2\leq d\leq n$ we get $0\leq k\leq n-2$. We recall that from [2, Theorem 9] we have

[TABLE]

Theorem 3.1

Let $q=2$ and write $d-1=k+1$. If $0<k<n-2$ or $k=n-2\geq 0$, then

[TABLE]

and if $k=0$ and $n\geq 3$ then

[TABLE]

**Proof: ** We start with the case where $k=0$ and $n\geq 3$. Let $f\in\mathbb{F}_{q}[X_{0},\ldots,X_{n}]_{d}$ be a nonzero polynomial such that

[TABLE]

and observe that

[TABLE]

Let $S$ be the support of $f$, then $S$ has property 1 of Proposition 2.6 and from Lemma 2.4 it also has property 2 so there exists a hyperplane $H\subset\mathbb{P}^{n}(\mathbb{F}_{q})$ such that $S\cap H=\emptyset$ and from Lemma 2.5 we must have $|f|=W^{(1)}_{\textrm{PRM}}(n,d)$. This shows that

[TABLE]

and let $g=X_{0}X_{3}+X_{1}X_{2}$. Let $\widetilde{H}_{n-2}$ be the linear subspace defined by $X_{0}=X_{1}=0$ and note that the support of $g$ does not meet $\widetilde{H}_{n-2}$. Counting the number of hyperplanes that contain $\widetilde{H}_{n-2}$ (as in the proof of Proposition 2.6) we get that a total of three hyperplanes, which we call $G_{0}$, $G_{1}$ and $G_{2}$, whose equations are, respectively, $X_{0}=0$, $X_{1}=0$ and $X_{0}+X_{1}=0$. Now it is easy to check that in each of these hyperplanes we have $2^{n-2}$ points in the support of $g$, hence $|g|=3\cdot 2^{n-2}$, which settles this case.

Assume now that $k$ is in the range $0<k<n-2$, and for $d=k+2$ let $f\in\mathbb{F}_{q}[X_{0},\ldots,X_{n}]_{d}$ be a nonzero polynomial such that

[TABLE]

(we observe that such a polynomial exists because $W^{(1)}_{\textrm{PRM}}(n,d)=2^{n-k-1}$). Let $S$ be the support of $f$, then $S$ has property 1 of Proposition 2.6 and from Lemma 2.4 it also has property 2 so there exists a hyperplane $H\subset\mathbb{P}^{n}(\mathbb{F}_{q})$ such that $S\cap H=\emptyset$. From Lemma 2.5 we have that either $|f|=W^{(1)}_{\textrm{PRM}}(n,d)$ or $|f|\geq W^{(2)}_{\textrm{RM}}(n,d-1)$. Note that $W^{(2)}_{\textrm{RM}}(n,d-1)=3\cdot 2^{n-k-2}=(1+1/2)2^{n-k-1}$, so we may conclude that $W^{(2)}_{\textrm{PRM}}(n,d)\geq W^{(2)}_{\textrm{RM}}(n,d-1)$, and we have already remarked in (2.2) that $W^{(2)}_{\textrm{PRM}}(n,d)\leq W^{(2)}_{\textrm{RM}}(n,d-1)$.

Now we assume that $k=n-2\geq 0$, and let $f\in\mathbb{F}_{q}[X_{0},\ldots,X_{n}]_{d}$ be a nonzero polynomial such that $0\neq|f|\leq 4$ (such polynomial exists because $W^{(1)}_{\textrm{PRM}}(n,d)=2$ in this case, where $d=n$). Then we may apply Proposition 2.7 and find that there exists a linear subspace $H_{r}$ of dimension $r\geq n-2$ such that $H_{r}\cap S=\emptyset$.

If $r=n-1$ then lemma 2.5 implies that $|f|=W^{(1)}_{\textrm{PRM}}(n,d)$ or $|f|\geq W^{(2)}_{\textrm{RM}}(n,d-1)$, and were done because from (2.2) we know that $W^{(2)}_{\textrm{PRM}}(n,d)\leq W^{(2)}_{\textrm{RM}}(n,d-1)$. Thus we assume that $r=n-2$ and after a projective transformation, if necessary, we assume that $H_{n-2}$ is the linear subspace defined by $X_{0}=X_{1}=0$. As above we have three hyperplanes which contain $H_{n-2}$, which we call $G_{0}$, $G_{1}$, $G_{2}$, and whose equations are, respectively, $X_{0}=0$, $X_{1}=0$ and $X_{0}+X_{1}=0$. Since $|S\cap G_{i}|\geq 1$ for $i=0,1,2$ we get that $|f|\geq 3>W^{(1)}_{\textrm{PRM}}(n,d)=2$ (recall that $d=n$). Thus $3\leq|f|\leq 4$ and we want to prove that $|f|=4=W^{(2)}_{\textrm{RM}}(n,d-1)$.

Let’s assume, by means of absurd, that $|f|=3$, so that $|S\cap G_{i}|=1$ for $i=0,1,2$ and let $(0:1:Q_{0})$, $(1:0:Q_{1})$ and $(1:1:Q_{2})$ be the points of intersection of $S$ with $G_{0}$, $G_{1}$ and $G_{2}$ respectively. Write $f$ as $f=X_{0}(X_{1}f_{2}+f_{1})+X_{1}f_{0}+f_{3}$, with $f_{2}\in\mathbb{F}_{q}[X_{0},\ldots,X_{n}]_{d-2}$, $f_{1}\in\mathbb{F}_{q}[X_{0},X_{2},\ldots,X_{n}]_{d-1}$, $f_{0}\in\mathbb{F}_{q}[X_{1},\ldots,X_{n}]_{d-1}$ and $f_{3}\in\mathbb{F}_{q}[X_{2},\ldots,X_{n}]_{d}$. Since $f$ vanishes on $H_{n-2}$ we get that $f_{3}$ vanishes on $H_{n-2}$ and a fortiori on $\mathbb{P}^{n}(\mathbb{F}_{q})$, so we may assume that $f=X_{0}(X_{1}f_{2}+f_{1})+X_{1}f_{0}$. From the definitions of $Q_{0}$, $Q_{1}$ and $Q_{2}$ we get that $f_{0}(1,Q_{0})=1$, $f_{1}(1,Q_{1})=1$ and

[TABLE]

Observe that $f_{2}(1,1,X_{2},\ldots,X_{n})$ is the zero polynomial or a polynomial of degree $d-2=n-2$ taking values on $\mathbb{A}^{n-1}(\mathbb{F}_{q})$, and in the latter case we have that either

[TABLE]

or

[TABLE]

We cannot have $f_{2}(1,1,X_{2},\ldots,X_{n})=0$ otherwise, if $Q_{0}\neq Q_{1}$ we would have $g(Q_{0})=f_{0}(1,Q_{0})=1$ and $g(Q_{1})=f_{1}(1,Q_{1})=1$, and if $Q_{0}=Q_{1}$ then $g(Q)=0$ for all $Q\in\mathbb{A}^{n-1}(\mathbb{F}_{q})$, so in both cases we have a contradiction with $|S\cap G_{2}|=1$. So we assume that $f_{2}(1,1,X_{2},\ldots,X_{n})\neq 0$ and in what follows we show that also in this case we cannot have $|S\cap G_{2}|=1$, which will conclude the proof that $|f|=4$. We split the proof in two parts.

I)Suppose that $f_{2}(1,1,Q_{2})=0$, from $g(Q_{2})=1$ we must have $Q_{2}=Q_{0}$ or $Q_{2}=Q_{1}$, and $Q_{0}\neq Q_{1}$. We will assume that $Q_{2}=Q_{0}$, the case where $Q_{2}=Q_{1}$ being similar. We know that there are at least two distinct points $Q_{3},Q_{4}\in\mathbb{A}^{n-1}(\mathbb{F}_{q})$ such that $f_{2}(1,1,Q_{i})=1$ for $i=3,4$. Clearly $Q_{3}\neq Q_{2}$, if $Q_{3}\neq Q_{1}$ we get $g(Q_{3})=1$ and if $Q_{3}=Q_{1}$ then $Q_{4}\neq Q_{1}$, $Q_{4}\neq Q_{2}=Q_{0}$ and $g(Q_{4})=1$, so $|S\cap G_{2}|>1$.

II) Now we assume that $f_{2}(1,1,Q_{2})=1$, from $g(Q_{2})=1$ we get that either $Q_{2}=Q_{0}=Q_{1}$, or $Q_{2}\neq Q_{0}$ and $Q_{2}\neq Q_{1}$. From $|f_{2}(1,1,X_{2},\ldots,X_{n})|\geq 2$ we know that there exists $Q_{3}\neq Q_{2}$ such that $f_{2}(1,1,Q_{3})=1$, so if $Q_{2}=Q_{0}=Q_{1}$ then $g(Q_{3})=1$ and $|S\cap G_{2}|>1$. Thus we assume now that $Q_{2}\neq Q_{0}$ and $Q_{2}\neq Q_{1}$. If $Q_{0}=Q_{1}$ then in both cases $Q_{3}=Q_{0}=Q_{1}$ or $Q_{3}\neq Q_{0}=Q_{1}$ we have $g(Q_{3})=1$. So now we consider the case where $Q_{0}\neq Q_{1}$. If $Q_{3}\notin\{Q_{0},Q_{1}\}$ then $g(Q_{3})=1$, if $Q_{3}\in\{Q_{0},Q_{1}\}$, say $Q_{3}=Q_{0}$ (the case where $Q_{3}=Q_{1}$ is similar) then $g(Q_{1})=1$ when $f_{2}(1,1,Q_{1})=0$, if $f_{2}(1,1,Q_{1})=1$ then we already have three distinct points of $\mathbb{A}^{n-1}(\mathbb{F}_{q})$ which are not zeros of $f_{2}(1,1,X_{2},\ldots,X_{n})$ (namely, $Q_{1}$, $Q_{2}$ and $Q_{3}$) so from inequality (3.1) above there is a point $\widetilde{Q}$, distinct from $Q_{1}$, $Q_{2}$ and $Q_{3}=Q_{0}$ such that $f_{2}(1,1,\widetilde{Q})=1$, hence $g(\widetilde{Q})=1$ and again $|S\cap G_{2}|>1$. This completes the proof of the case $k=n-2$ and of the Theorem. $\Box$

In [4] Delsarte et al. proved that the codewords of minimal weight in $\textrm{RM}(n,d)$ are such that their support is the union of certain affine subspaces of $\mathbb{F}_{q}^{q^{n}}$, or equivalently, that these codewords may be obtained as the evaluation of polynomials whose classes in $\mathbb{F}_{q}[X_{1},\ldots,X_{n}]/I_{q}$ may be represented by the product of $d$ polynomials of degree 1. In [7] the author proves a similar result for the next-to-minimal codewords of $\textrm{RM}(n,d)$, in the case where $q\geq 3$. In [14] (see also [2]) the author proves that also for $\textrm{PRM}(n,d)$ the minimal weight codewords may be characterized as being the evaluation of certain homogeneous polynomials whose classes in $\mathbb{F}_{q}[X_{0},\ldots,X_{n}]/J_{q}$ can be written as the product of linear factors, so that the zeros of such polynomials are over a union of hyperplanes. As a byproduct of the above proof we see that, for $q=2$, such statement is not true for the support of the next-to-minimal codewords, since in the case where $k=0$ and $n\geq 3$ we presented a codeword whose zeros form an irreducible quadric in $\mathbb{P}^{n}(\mathbb{F}_{q})$.